Binary Linear Codes and Binary Matrices

Binary Linear Block Codes

New Binary Linear Codes

Capacity Achieving Linear Codes with Random Binary Sparse Generating Matrices

In this paper, we prove the existence of capacity achieving linear codes with random binary sparse generating matrices. The results on the existence of capacity achieving linear codes in the literature are limited to the random binary codes with equal probability generating matrix elements and sparse parity-check matrices. Moreover, the codes with sparse generating matrices reported in the lite...

Binary Codes and Kasteleyn 3-matrices

Abstract. Two cornerstones of the Kasteleyn method are: 1. rewriting the Ising partition function as the dimer partition function, that is, the generating function of the perfect matchings, and 2. expressing the dimer partition function of planar graphs as the determinant. This paper initiates the 3-dimensional Kasteleyn method. We show that the weight enumerator of any binary linear code is po...

Binary and Ternary Codes of Hadamard Matrices∗

Let H be a Hadamard matrix of order 24. In this note, we show that the extremality of the binary code of H is equivalent to the extremality of the ternary code of HT . This fact has been observed by Assmus and Key [1], as a result of the complete classification of Hadamard matrices of order 24. Our proof is a consequence of more general results on the minimum weight of the dual code of the code...